If one drops the strong assumption that firms and households know all of the relevant parameters, and instead models agents as learning these parameters, the estimated parameters become random variables. Taking expectations several periods into the future may then involve taking the expectation of a product of random variables. Because the resulting problem is difficult, previous research has avoided it. This paper makes some progress using both analytical and numerical techniques. Focusing especially on consumption, we find that the resulting covariance terms could account for the well-documented empirical result that consumption displays excess sensitivity to lagged income. We show that a similar covariance effect could play a role in many widely used economic models.